The Chaos Within Sudoku

TL;DR

This paper models Sudoku puzzles as a dynamical system and demonstrates that the difficulty correlates with transient chaos, allowing a quantitative hardness scale based on escape rates.

Contribution

It introduces a novel dynamical systems approach to quantify Sudoku difficulty through transient chaos and escape rates, creating a new hardness scale.

Findings

Escape rate correlates with human difficulty ratings.

A 'Richter'-type scale for Sudoku hardness is proposed.

No puzzles with hardness above a certain threshold are known.

Abstract

The mathematical structure of the widely popular Sudoku puzzles is akin to typical hard constraint satisfaction problems that lie at the heart of many applications, including protein folding and the general problem of finding the ground state of a glassy spin system. Via an exact mapping of Sudoku into a deterministic, continuous-time dynamical system, here we show that the difficulty of Sudoku translates into transient chaotic behavior exhibited by the dynamical system. In particular, we show that the escape rate $\kappa$, an invariant characteristic of transient chaos, provides a single scalar measure of the puzzle's hardness, which correlates well with human difficulty level ratings. Accordingly, $\eta = -\log_{10}{\kappa}$ can be used to define a "Richter"-type scale for puzzle hardness, with easy puzzles falling in the range $0 < \eta \leq 1$, medium ones within $1 < \eta \leq 2$, hard in $2 < \eta \leq 3$ and ultra-hard with $\eta > 3$. To our best knowledge, there are no known puzzles with $\eta > 4$.